Comparative biology and PGLS

Natalie Cooper (natalie.cooper@nhm.ac.uk)

Feb 2017

Philosophical approach to answering biological questions

Are large animals more likely to become extinct than small animals?

  1. Choose one species (e.g. tiger) and find out everything you can about it, and why it is going extinct.
    • really detailed answer
    • BUT only applies to tigers
    • no replication (N = 1)

Philosophical approach to answering biological questions

Are large animals more likely to become extinct than small animals?

  1. Investigate multiple species and look for correlations among their extinction risk and their traits.
    • broadly applicable answer
    • BUT lacks detail, ignores outliers

Ideal is probably somewhere in between…

The comparative method

These are all comparative questions that we can test by comparing different species.

This is the classical way people studied biodiversity - e.g. Darwin, Wallace etc.

Comparative methods and phylogenies

What can we do with phylogenies and PCMs?

Phylogenetic non-independence/pseudoreplication

Phylogenetic non-independence/pseudoreplication

When you fit a linear regression you assume your data points are independent…

Phylogenetic non-independence/pseudoreplication

But if your points are species they are not…instead you find close relatives often cluster together.

Phylogenetic non-independence/pseudoreplication

Phylogenetic non-independence/pseudoreplication

Phylogenetic non-independence/pseudoreplication

If you treat gorillas and chimps as independent, you count all the evolution which occurred on the red branch twice etc.

How do we deal with this problem?

Two main methods

A quick revision of OLS

A quick revision of OLS

\(Y = X \beta + \epsilon\)
\(height = femur length \beta + \epsilon\)

A quick revision of OLS

Residual errors should have:

A quick revision of OLS

Normal distribution (univariate)

A quick revision of OLS

Multivariate normal distribution

A quick revision of OLS

Generalised Least Squares (GLS)

Variance-covariance matrices in PGLS

Variance-covariance matrices in PGLS

Tree transforms

Pagel’s \(\lambda\)

\(\lambda\) transforms the tree by multiplying the off-diagonal elements (internal/shared branches) of the vcv matrix by \(\lambda\)

Pagel’s \(\lambda\) = 1

Tree remains the same = trait is evolving under Brownian motion (BM)

This is the same as independent contrasts

Pagel’s \(\lambda\) = 0

Tree collapses to a star phylogeny = trait is not related to phylogeny

This is the same as an OLS model

Pagel’s \(\lambda\) = 0.5

Internal branches get shorter = there is phylogenetic signal in the trait but not as much as under BM.

Pagel’s \(\lambda\) summary

Phylogenetic signal

Pattern where related species resemble each other more than more distant relatives

Note this is a pattern not a process (Kamilar & Cooper 2013; Losos 2011)

Phylogenetic signal

Two most used methods are:

  1. Pagel’s \(\lambda\) (Pagel 1997/99)
  2. Blomberg’s K (K not kappa) (Blomberg et al 2003)

Blomberg’s K

Blomberg’s K

Blomberg’s K

Blomberg’s K

Blomberg’s K

Blomberg’s K summary

Pagel’s \(\lambda\) versus Blomberg’s K

Brief caveat!!!